Mathematical Sciences: Analysis of the Quasi-Regular Representation of Lie Groups and Index Theory on Non-CompactManifolds

现代科学最基本的发展之一 就是量子力学理论 试图理解 物理学和基础数学都直接导致了 现代分析的两个重要领域的前沿， 表示论和算子代数。 表示 理论是一种利用固有的物理对称性的手段， 而算子代数的概念是为了提供 量化的正确框架。 最近有 我们对数学的理解大大扩展了 潜在的量化。 这来自于 物理概念和数学构造之间的联系 在 特别是表示论和算子代数， 最近聚集在一起形成了一个强大的理论， Atiyah-Singer指数定理。 福克斯教授是一位年轻的研究员， 数学的范围内必要的贡献这一领域。 他 是表示论、算子代数和 指数理论 他提出了一个指数定理， 非紧流形，推广了Connes的推广， 紧流形的Atiyah-Singer指标定理 此索引 定理将被用来研究离散谱的 半单李群的拟正则表示 有限体积的局部对称空间。 除了其 表示论与算子内在兴趣 代数，这个问题的解决方案将导致更好的 理解量子化的各个方面。 的 调查人员先前的工作，部分是合作，奠定了一个 为开展这项研究奠定了坚实的基础。

One of the most fundamental developments in modern science has been the theory of quantum mechanics. The attempt to understand both the physics and the underlying mathematics has led directly to current frontiers in two important areas of Modern Analysis, representation theory and operator algebras. Representation theory is a means of exploiting the inherent physical symmetries, while the concept of operator algebras was invented to provide the correct framework for quantization. Recently, there has been a great expansion in our understanding of the mathematics underlying quantization. This has come from deep interactions between physical ideas and mathematical constructs. In particular, representation theory and operator algebras have been brought together recently in a powerful theory that centers around the celebrated Atiyah-Singer index theorem. Professor Fox is a young researcher who has mastered the broad range of mathematics necessary to contribute to this field. He is an expert in representation theory, operator algebras, and index theory. He proposes to develop an index theorem for noncompact manifolds that generalizes Connes' extension of the Atiyah-Singer index theorem for compact manifolds. This index theorem would be used to study the discrete spectrum of the quasi-regular representation of a semi-simple Lie group acting on a locally symmetric space of finite volume. In addition to its intrinsic interest in representation theory and operator algebras, a solution to this problem would lead to a better understanding of the various facets of quantization. The investigator's prior work, in part collaborative, has laid a solid foundation for undertaking this study.